How do I animate a smooth transition between all colours in the RGB spectrum?
I believe that's the setup you wanted. By adjusting/animating value node you can achieve effect close enough to what I've seen in the video (or you can just use 'Hue Saturation Value' node input itself). You can adjust look of the gradient by changing texture scale (in the Mapping node), texture interpolation (in texture node itself) and by adjusting colors in the color ramp (for better look I recommend you to use HSV mode (with any sub-option) in the color ramp). 



Have a good day!